Research of Francisco J. Muñoz Almaraz

Introduction

We have studied (in a work joint with E. Doedel, A. Vanderbauwhede, E. Freire and J.Galán) about the continuation of periodic orbits in conservative systems.

Papers in journals

The main theoretical results are in the following paper

Continuation of periodic orbits in conservative and Hamiltonian systems. .
Physica D: Volume 181, Issues 1, Pages 1-38 (1 July 2003)
Pages 1-38
F. J. Muñoz-Almaraz, E. Freire, J. Galán, E. Doedel and A. Vanderbauwhede

These results allow us to use AUTO for the continuation in conservative systems.
We have applied these techniques in the continuation for the full three body problem:

Stability and Bifurcations of the Figure-8 Solution of the Three-Body Problem. .
Physical Review Letters: Volume 88 Paper: 241101

Similar ideas are applied for systems with time-reversal symmetries and first integrals. The theorical results and some families of periodic orbits are shown in the following paper which is going to appear in "Revista de la Real Academia de Ciencias de Zaragoza"

Families of symmetric periodic orbits in the three body problem and the figure eight
A complete paper with our research about time-reversal orbits has been accepted in Celestial Mechanichs
Preprint about time-reversal symmetries submitted to Celestial Mechanics
Most of these results are written in Spanish in my PhD Thesis

AUTO files

AUTO files to calculate periodic orbits from figure eight solution are supplied here.
AUTO files for the TBP
AUTO files to calculate periodic orbits with continuation from the Gerver's solution (or supereight):
AUTO files for four body problem

Movies

some movies (codecs are needed in Windows. They can be downloaded from DivX ) with these periodic orbits.

From figure eight

Figure eight Figure eight
(m1=m2=m3=1)

Family starting from Figure eight Family starting from Figure eight
(m1=0.6 ;m2=m3=1) (m1=0.94 ;m2=m3=1)

Family Family
(m1= 0.94;m2=m3=1) (m1=0.6 ;m2=m3=1)

Family Family
(m1= 1.005;m2=m3=1) (m1= 0.6;m2=m3=1)

from Gerver's solution


Family Family
Family Family

Some talks and poster:

Equadiff99 Numerical continuation of periodic orbits in symmetric hamiltonian systems
Cedya99 Continuación Numérica de Órbitas Periódicas en Sistemas Hamiltonianos con Simetría. (Spanish)
Cedya2001 Continuación de órbitas periódicas en sistemas hamiltonianos con simetría: aplicación a mecánica celeste. (Spanish)
WORKSHOP ON ANALYSIS AND CONTINUATION OF BIFURCATIONS Families of symmetric periodic orbits in the three body problem and the figure eight.
Poster in Summer Workshop on Advanced Topics in Astrodynamics Barcelona, July 5-10, 2004
Talk in CELMEC IV Cimino al San Martino (Viterbo), September 12-16, 2005
Talk in IX Jornadas de Trabajo en Mecánica Celeste Jaca (Huesca) June 26--28, 2006 Link with data for solutions
Francisco Javier Muñoz Almaraz